Limiting Logins to A Google ID

I am building a site for a small college that uses Google Apps as a “unified login,” as well as for some specific apps. I’d like to restrict certain areas of the site so only users who have logged into the site can get to them –but I’d like the login to be tied to the Google Apps username they receive as part of the College’s work.

The plugins that work in this space are more for “social,” work, meaning signing in with any google account allows the reader to make comments –but this really isn’t my requirement. I did see https://www.ads-software.com/support/plugin/google-identity-toolkit, but I’m not certain this would actually restrict users to one specific set of google identities…

Does anyone know of a plugin that’s specific built to provide this type of usage?

Thanks!